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On the Hausdorff dimension of general Cantor sets

Published online by Cambridge University Press:  24 October 2008

A. F. Beardon
A. F. Beardon
Affiliation:
Imperial College, London
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Extract

Introduction and notation. In this paper a generalization of the Cantor set is discussed. Upper and lower estimates of the Hausdorff dimension of such a set are obtained and, in particular, it is shown that the Hausdorff dimension is always positive and less than that of the underlying space. The concept of local dimension at a point is introduced and studied as a function of that point.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 3 , July 1965 , pp. 679 - 694
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

(1)Ford, L. R.Automorphic functions (Chelsea, 1951).Google Scholar
(2)Good, I. J.The fractional dimensional theory of continued fractions. Proc. Cambridge Philos. Soc. 37 (1941), 199–228.Google Scholar
(3)Hausdorff, F.Dimension und äusseres Mass. Math. Ann. 79 (1919), 157–179.CrossRefGoogle Scholar
(4)Moran, P. A. P.Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc. 42 (1946), 15–23.Google Scholar
(5)Rogers, C. A. and Taylor, S. J.Additive set functions in Euclidean space. Acta Math. 101 (1959), 273–302.Google Scholar
(6)Taylor, S. J.On the connexion between Hausdorff measures and generalized capacities. Proc. Cambridge Philos. Soc. 57 (1961) 524–531.Google Scholar
(7)Tsuji, M.On the capacity of general Cantor sets. J. Math. Soc. Japan 5 (1953), 235–252.CrossRefGoogle Scholar
(8)Tsuji, M.Potential theory in modern function theory (Tokyo, 1959).Google Scholar